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Truth values on generalizations of some commutative fuzzy structures
Fuzzy Sets and Systems 157 (2006) 3159 – 3168 www.elsevier.com/locate/fss
Truth values on generalizations of some commutative fuzzy structures Jiˇrí Rach˚uneka,∗,1 , Dana Šalounováb a Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic b Department of Mathematical Methods in Economy, Faculty of Economics, VŠB—Technical University Ostrava, Sokolská 33,
701 21 Ostrava, Czech Republic Received 1 December 2005; received in revised form 24 August 2006; accepted 30 August 2006 Available online 25 September 2006
Abstract Hájek introduced the logic BLvt enriching the logic BL by a unary connective vt which is a formalization of Zadeh’s fuzzy truth value “very true”. BLvt -algebras, i.e. BL-algebras with unary operations, called vt-operators, which are among others subdiagonal, are an algebraic counterpart of BLvt . Residuated lattice ordered monoids (R-monoids) are common generalizations of BL-algebras and Heyting algebras. In the paper, we study algebraic properties of Rvt -algebras (and consequently of BLvt -algebras) and of those that are enriched by derived superdiagonal operators which in the case of MV-algebras are the duals to vt-operators. © 2006 Elsevier B.V. All rights reserved. MSC: 03B47; 03B52; 03G25; 06D35; 06F05 Keywords: Residuated -monoid; Basic fuzzy logic; BL-algebra; Fuzzy truth value
1. Introduction Inspired by the considerations of Zadeh [33], Hájek in [16] formalized the fuzzy truth value “very true”. He enriched the language of the basic fuzzy logic BL by adding a new unary connective vt and introduced the (propositional) logic BL vt . The completeness of BL vt was proved in [16] by using the so-called BL vt -algebras, an algebraic counterpart of BL vt . Recall that a BL vt -algebra is a BL-algebra with a unary operation (an operator) which is, among others, subdiagonal. Bounded commutative residuated lattice ordered monoids (R-monoids) are algebraic structures which generalize, e.g. both BL-algebras and Heyting algebras (an algebraic counterpart of the intuitionistic propositional logic). Nevertheless, many of properties of BL-algebras are also satisfied in any bounded commutative R-monoids. Therefore, bounded commutative R-monoids could be taken as an algebraic semantics of a more general logic than Hájek’s fuzzy logic. It is known that every BL-algebra (and consequently every MV-algebra) can be represented as a subdirect product of linearly ordered BL-algebras. (Recall also that MV-algebras are categorically isomorphic to Wajsberg algebras [10].) ∗ Corresponding author. Tel.: +420 585 634 650; fax: +420 585 634 663.
E-mail addresses: [email protected] (J. Rach˚unek), [email protected] (D. Šalounová). 1 Supported by the Council of Czech Government, MSM 6198959214.
0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.08.007
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It is also true [26] that a bounded commutative R-monoid is a subdirect product of linearly ordered R-monoids if and only if it is a BL-algebra. On the other side, R-monoids which need not be BL-algebras can be constructed from BL-algebras by means of other natural operations, e.g. by means of pastings (= ordinal sums). Note that by [18,19], the pasting of linearly ordered Wajsberg algebras is a linearly ordered BL-algebra, but the pasting of Wajsberg algebras which are not linear, gives bounded commutative R-monoids which are not BL-algebras. Therefore, by a “commutative fuzzy structure” we mean here any BL-algebra and as its appropriate generalization is considered any bounded commutative R-monoid. In this paper we introduce Rvt -monoids as bounded commutative R-monoids with a unary subdiagonal and monotone self-mapping (called a vt-operator) which generalize BL vt -algebras. Further, we derive from the vt-operators, in a purely algebraic and uniform way, superdiagonal and monotone self-mappings which are in the case of MV-algebras vf -operators, i.e. formalizations of the fuzzy truth value “very false”. Note that in the case of BL-algebras, vt-operators induce so-called st-operators that have been introduced very recently by Vychodil [31] as formalization of the fuzzy truth value “slightly true”. Note, nevertheless, that these superdiagonal operators are the duals to vt-operators only in the case of MV-algebras. For concepts and results concerning MV-algebras, BL-algebras and Heyting algebras see for instance [4,15,2]. 2. Bounded commutative R-monoids Bounded commutative R-monoids form a large class of algebras containing as proper subclasses, among others, the classes of algebras of some logics, e.g. the class of BL-algebras, i.e. algebras of the basic fuzzy logic (and consequently the class of MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic), as well as the class of Heyting algebras, i.e. algebras of the intuitionistic logic. Bounded commutative R-monoids have many common properties of both mentioned kinds of algebras of logics and hence, they could be taken as the algebras of a logic which generalizes fuzzy logics and admits also the intuitionistic principles. Definition 1. A bounded commutative R-monoid is an algebra M = (M; ∨, ∧, , →, 0, 1) of type 2, 2, 2, 2, 0, 0 satisfying the following conditions: (i) (ii) (iii) (iv)
(M; , 1) is a commutative monoid. (M; ∨, ∧, 0, 1) is a bounded lattice. x y z if and only if x y → z for any x, y, z ∈ M. x (x → y) = x ∧ y for any x, y ∈ M.
Remark 2. In fact, the notion of a bounded commutative R-monoid is a duplicate name for a commutative residuated lattice [32,24] satisfying divisibility condition or for a divisible commutative residuated lattice [18], which is categorically isomorphic to a divisible BCK(P) lattice [18], or for a divisible integral residuated commutative -monoid [17] or for a bounded commutative integral generalized BL-algebra [3,22]. In the sequel, by an R-monoid we will mean a bounded commutative R-monoid. It is possible to show that for any R-monoid M, the lattice (M; ∨, ∧) is distributive and that operation distributes over the lattice operations ∨ and ∧. Let us define on any R-monoid M the unary operation − (negation) by x − := x → 0. Algebras of the above mentioned propositional logics can be characterized in the class of all R-monoids as follows: An R-monoid M is (a) a BL-algebra [26] if and only if M satisfies the identity of pre-linearity (x → y) ∨ (y → x) = 1; (b) an MV-algebra [17], see also [9], if and only if M fulfils the double negation law x −− = x; (c) a Heyting algebra [30] if and only if the operations and ∧ coincide on M.
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Remark 3. Let us recall that commutative residuated lattices which satisfy the identity of pre-linearity are also called MTL-algebras and by [8,6] they are an algebraic counterpart of the so-called Monoidal t-norm logic (MTL, for short). Now it is obvious that an R-monoid is an MTL-algebra if and only if it is a BL-algebra. The facts in the rest of the section can be verified as for the BL case, since the pre-linearity condition is not involved, and hence the proofs are omitted. Lemma 4. In any bounded commutative R-monoid M we have for any x, y, z ∈ M: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
x y ⇐⇒ x → y = 1. x y ⇒ x z y z. x y ⇒ z → x z → y, y → z x → z. x → x = 1, 1 → x = x, x → 1 = 1. y x → y. x x −− , x − = x −−− . x y ⇒ y − x − . (x ∨ y)− = x − ∧ y − . (x ∧ y)−− = x −− ∧ y −− . (x y)− = y → x − = y −− → x − = x → y − = x −− → y − . (x → y)−− = x −− → y −− .
If M is an R-monoid and ∅ = F ⊆ M, then F is called a filter of M if (i) x, y ∈ F ⇒ x y ∈ F ; (ii) x ∈ F, y ∈ M, x y ⇒ y ∈ F . The filters of R-monoids coincide with the kernels of their congruences. If F is a filter of M then F is the kernel of the unique congruence (F ) such that x, y ∈ (F ) if and only if (x → y) ∧ (y → x) ∈ F , for any x, y ∈ M. Hence, we will consider quotient R-monoids M/F of R-monoid M with respect to its filters F . 3. R-monoids with vt-operators In this section we deal with Rvt -monoids which are generalizations of BL vt -algebras introduced in [16]. Definition 5. (a) Let M be an R-monoid. A mapping v : M −→ M is called a weak vt-operator (wvt-operator in brief) on M if for any x, y ∈ M: (1) v(1) = 1, (2) v(x)x, i.e. v is subdiagonal, (3) v(x → y) v(x) → v(y). (b) If a wvt-operator v satisfies for any x, y ∈ M (4) v(x ∨ y)v(x) ∨ v(y), then v is called a vt-operator on M. Remark 6. Any R-monoid admits vt-operators, e.g. the identity and the globalization g, where g(x) = 0 for x = 1 and g(1) = 1. Lemma 7. Let v be a weak vt-operator on an R-monoid and x, y, z ∈ M. Then (a) (b) (c) (d)
v(0) = 0, x y ⇒ v(x)v(y), v(x − )(v(x))− , x y z ⇒ v(x) v(y)v(z),
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(e) v(x) v(y)v(x y), (f) v(x) v(x → y)v(x ∧ y)v(x) ∧ v(y). If v is a vt-operator on M and x, y ∈ M then (g) v(x ∨ y) = v(x) ∨ v(y). Proof. (a) By the definition, v(0) 0, hence v(0) = 0. (b) Let x, y ∈ M and x y. Then x → y = 1, hence by (3) and (1) from definition, we get v(x) → v(y) = 1, and thus v(x)v(y). (c) Let x ∈ M. Then by (3) from definition and by (a), v(x − ) = v(x → 0) v(x) → v(0) = v(x) → 0 = (v(x))− . (d) Let x y z. Then x y → z, so by (b) and (3), v(x) v(y → z) v(y) → v(z), and from this, v(x) v(y)v(z). (e) It follows from (d) for z = x y. (f) By (e) and (b), v(x) v(x → y)v(x (x → y)) = v(x ∧ y) v(x) ∧ v(y). (g) By (b), we have v(x) ∨ v(y)v(x ∨ y), hence by (4), v(x ∨ y) = v(x) ∨ v(y). Definition 8. If M is an R-monoid and v is a vt-operator (a wvt-operator) on M then the ordered pair (M, v) is called an Rvt -monoid (a Rwvt -monoid, respectively). Remark 9. (a) For the sake of simplicity, we will mainly study properties of Rvt -monoids and some operators derived from them. (b) If (M, v) is an Rvt -monoid and M is a BL-algebra then (M, v) is a BL vt -algebra in the sense of [16]. Definition 10. If (M, v) is an Rvt -monoid and F is a filter of M then F is called a v-filter of the Rvt -monoid (M, v) if v(x) ∈ F for every x ∈ F . For any v-filter F of an Rvt -monoid (M, v) denote by vF : M/F −→ M/F the mapping such that vF (x/F ) := v(x)/F, for each x ∈ M. Proposition 11. If F is a v-filter of an Rvt -monoid (M, v) then vF is a vt-operator on the quotient Rmonoid M/F . Proof. Firstly we will show that vF is a correctly defined mapping of M/F into M/F . Let x, y ∈ M and x/F = y/F . Then x, y ∈ (F ), i.e. (x → y) ∧ (y → x) ∈ F, and thus also x → y, y → x ∈ F. Since F is a v-filter, we get v(x → y), v(y → x) ∈ F, and hence by (3) from the definition of a vt-operator we obtain v(x) → v(y), v(y) → v(x) ∈ F. By Lemma 4(5), v(y) → v(x)(v(x) → v(y)) → (v(y) → v(x)), hence (v(x) → v(y)) → (v(y) → v(x)) ∈ F, and this means v(x), v(y) ∈ (F ), i.e. vF (x/F ) = vF (y/F ). Now it is easy to verify that the mapping vF is a vt-operator on M/F. (1) vF (1/F ) = v(1)/F = 1/F. (2) vF (x/F ) = v(x)/F x/F. (3) vF (x/F → y/F ) = vF ((x → y)/F ) = (v(x → y))/F (v(x) → v(y))/F = v(x)/F → v(y)/F = vF (x/F ) → vF (y/F ). (4) vF (x/F ∨ y/F ) = vF ((x ∨ y)/F ) = v(x ∨ y)/F (v(x) ∨ v(y))/F = (v(x)/F ) ∨ (v(y)/F ) = vF (x/F ) ∨ vF (y/F ). Definition 12. If (M, v) is an Rvt -monoid and is a congruence on the R-monoid M then is called a v-congruence on (M, v), if x, y ∈ implies v(x), v(y) ∈ for each x, y ∈ M. Proposition 13. If (M, v) is an Rvt -monoid then there is a one-to-one correspondence between its v-filters and v-congruences.
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Proof. (a) Let be a v-congruence on (M, v) and let F = 1/F = {x ∈ M : x, 1 ∈ }. Then F is a filter of the R-monoid M. Let us suppose that x ∈ F . Then x, 1 ∈ , hence v(x), 1 = v(x), v(1) ∈ , and therefore v(x) ∈ F . That means F is a v-filter on (M, v). (b) Let F be a v-filter of (M, v) and let F be the corresponding congruence on M, i.e. x, y ∈ F if and only if (x → y) ∧ (y → x) ∈ F. Hence, if x, y ∈ F then also v((x → y) ∧ (y → x)) ∈ F . Let x, y ∈ F . Then by property (3) of a vt-operator and Lemma 7(f), (v(x) → v(y)) ∧ ((v(y) → v(x))v(x → y) ∧ v(y → x)v((x → y) ∧ (y → x)) ∈ F, hence (v(x) → v(y)) ∧ ((v(y) → v(x)) ∈ F, and this means v(x), v(y) ∈ F . Therefore F is a v-congruence on (M, v). Now we will deal with Rwvt -monoids (M, v) satisfying the identity (5) v(x → y) ∨ v(y → x) = 1. Theorem 14. If M is an R-monoid then there is a wvt-operator v on M satisfying (5) if and only if M is a BL-algebra. Proof. (a) Let M be an R-monoid which is not a BL-algebra. Then there exist x, y ∈ M such that (x → y) ∨ (y → x) = 1. Hence for any wvt-operator v on M, v(x → y) ∨ v(y → x)(x → y) ∨ (y → x) < 1, therefore (5) fails. (b) Let M be a BL-algebra. Then for each vt-operator v on M and x, y ∈ M, v(x → y) ∨ v(y → x) = v((x → y) ∨ (y → x)) = v(1) = 1, thus M satisfies (5).
An Rvt -monoid (M, v) is called an Rvt -chain if the R-monoid M is linearly ordered. By [26], the class of BLalgebras coincides with the class of (bounded commutative) R-monoids which are representable as subdirect products of R-chains. Hence, among others, every Rvt -chain is in fact a BL vt -chain. We will prove that every BL vt -algebra is a subdirect product not only of BL-chains (i.e., as a BL-algebra in the corresponding signature), but, moreover, it is also such a product of BL vt -chains (in the extended signature). Recall that a filter P of an R-monoid M is called prime if P = F ∩ G implies P = F or P = G for any filters F and G of M. A prime filter is called minimal if it is a minimal element in the sets of prime filters of M ordered by set inclusion. By Zorn’s lemma, every prime filter of M contains a minimal prime filter. For any a ∈ M put a ⊥ = {x ∈ M : x ∨ a = 1}. If P is a prime filter of M then x ∨ y = 1 implies x ∈ P or y ∈ P for each x, y ∈ M and then the quotient R-monoid M/P is linearly ordered. If P is a minimal prime filter of M then P = {a ⊥ : a ∈ M \ P }. (See e.g. [25].) Theorem 15. Every BL vt -algebra is a subdirect product of BL vt -chains. Proof. It is obvious that it suffices to prove that every BL vt -algebra is isomorphic to subdirect product of BL vt -chains. Since any BL-algebra M is representable as a subdirect product of BL-chains, the intersection of all minimal prime filters of M is equal to {1}. Hence, it remains to show that every minimal prime filter of M is a v-filter. Let P be a minimal prime filter of M. Then P = {a ⊥ : a ∈ M \ P }. Let x ∈ P . Then there exists a ∈ / P such that x ∨ a = 1, hence 1 = v(1) = v(x ∨ a) = v(x) ∨ v(a). Since a ∈ / P , we get v(a) ∈ / P , therefore (P is a prime filter) v(x) ∈ P . 4. Operators on R-monoids derived from vt-operators Every vt-operator on an R-monoid M is, by the definition and Lemma 7(b), a subdiagonal and monotone selfmapping of M. Now, we use vt-operators to introduce derived self-mappings of M that are, among others, superdiagonal and monotone, and in the case of MV-algebras they have the properties of unary connectives “very false”.
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If M is a R-monoid and f : M −→ M then we denote by f − the mapping of M into M such that for any x ∈ M, f − (x) = (f (x − ))− . Example 16. Let us consider the standard MV-algebra [0, 1] = (R, 1). It is known that the mapping v : [0, 1] − → [0, 1] such that v(x) = x 2 is a vt-operator on [0, 1]. Then v − : [0, 1] −→ [0, 1] is the mapping such that v − (x) = 2x − x 2 for each x ∈ [0, 1]. We say that an R-monoid M is normal if M satisfies the identity (x y)−− = x −− y −− . Remark 17. Every BL-algebra and every Heyting algebra is normal (see [28]), hence the variety of normal R-monoids is considerably wide. Proposition 18. Let (M, v) be an Rvt -monoid. Then we have for any x, y, z ∈ M: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
v − (0) = 0, v − (1) = 1, x v − (x), x y implies v − (x)v − (y), v − (x ∧ y) v − (x) ∧ v − (y), v − (x ∨ y) v − (x) ∨ v − (y), v − (x → y −− )v(x) → v − (y), x y z implies v − (x) v(y)v − (z), v − (x) v(y)v − (x y), v − (x) v(x → y) v − (x ∧ y). If M is normal then v − (x → y)v(x −− ) → v − (y) v(x) → v − (y). If M is an MV-algebra then v − (x → y)v(x) → v − (y).
Proof. 1. v − (0) = (v(0− ))− = 1− = 0, v − (1) = (v(1− ))− = (v(0))− = 0− = 1. 2. v − (x) = ((v(x − ))− x −− x. 3. x y ⇒ x − y − ⇒ v(x − ) v(y − ) ⇒ (v(x − ))− (v(y − ))− ⇒ v − (x) v − (y). 4. and 5. They follow from 3. 6. Using Lemma 4(6), (7), (12) and Lemma 7(a), (c), we have v − (x → y −− ) = v − ((xy − )− ) = (v((xy − )−− ))− , v(x) → v − (y) = v(x) → (v(y − ))− = ((v(x) v(y − ))− (v(x y − ))− (v((x y − )−− )− , hence v − (x → y −− )v(x) → v − (y). 7. By 3 and 6 and by Lemma 4(3) we get x y z ⇒ x y → z ⇒ v − (x)v − (y → z)v − (y → z−− ) ⇒ v − (x)v(y) → v − (z) ⇒ v − (x) v(y) v − (z). 8. It follows from 7. 9. v − (x) v(x → y)v − (x (x → y)) = v − (x ∧ y). 10. By [27, Lemma 2.3], x → y y − → x − , thus from the normality of M we get v − (x → y) = (v((x → y)− ))− (v((y − → x − )− ))− = (v((y − x)−− ))− = (v(y − x −− ))− . Furthermore, (v(y − ) v(x −− ))− = v(x −− ) → (v(y − ))− = v(x −− ) → v − (y), and since by Lemma 7(d), (v(y − x −− ))− (v(y − ) v(x −− ))− , we obtain v − (x → y) v(x −− ) → v − (y). 11. It follows from 6 as well as from 10. If M is an MV-algebra then M satisfies the double negation law x −− = x, hence there exist on M both a t-norm and its associated residuum → and a t-conorm and its associated residuum, say . (See [11–13,23,18].)
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Consequently, on any MV-algebra M one can define not only vt-operators but also dual operators, vf -operators (vf = very false). We will show that every vt-operator on M determines a vf -operator on M.As usual, put x y := (x − y − )− and x y := x y − , for any x, y ∈ M. Proposition 19. If M is an MV-algebra and v is a vt-operator on M then v − is a vf -operator on M, i.e. for each x, y ∈ M it holds: (1− ) (2− ) (3− ) (4− )
v − (0) = 0, v − (x)x, v − (x) v − (y)v − (x y), v − (x ∧ y) = v − (x) ∧ v − (y).
Proof. It remains to verify properties (3− ) and (4− ). (3− ): Using 8 from Proposition 18, we get v − (x) v − (y) = (v(x − )− (v(y − ))− = (v(x − ))− (v(y − ))−− = (v(x − ))− v(y − ) = v − (x) v(y − ) v − (x y − ) = v − (x y). (4− ): Since any MV-algebra satisfies de Morgan laws for the lattice operations, we have v − (x ∧ y) = (v((x ∧ y)− ))− = (v(x − ∨ y − ))− = (v(x − ) ∨ v(y − ))− = (v(x − ))− ∧ (v(y − ))− = v − (x) ∧ v − (y). Remark 20. Very recently, Vychodil [31] has introduced the notion of a BL vt,st -algebra in order to study the socalled truth depressing hedges on BL-algebras. An algebra (M; ∨, ∧, , →, v, s, 0, 1) is called a BL vt,st -algebra if M = (M; ∨, ∧, , →, 0, 1) is a BL-algebra, v is a wvt-operator on M and s : M −→ M is a mapping such that () s(0) = 0; () x s(x); () v(x → y)s(x) → s(y). We will show that if M is a BL-algebra and v is a wvt-operator on M then (M; ∨, ∧, , →, v, v − , 0, 1) is a BL vt,st algebra. Let x, y ∈ M. Then, by Lemma 4(8), v − (x) → v − (y) = (v − (x − ))− → (v(y − ))− v(y − ) → v(x − ) v(y − → x − )v(x → y). Definition 21. Let M be an R-monoid. A mapping w : M −→ M is called a near vt-operator on M if it satisfies conditions (1), (3) and (4) from the definition of a vt-operator on M and if for any x ∈ M, (2 ) w(x)x −− . Remark 22. By Lemma 4(6), every vt-operator on M is a near vt-operator on M. Let M be an R-monoid and f : M −→ M be a mapping. Let us denote by f˜ the self-mapping of M such that ˜ f = (f − )− , i.e., for each x ∈ M, f˜(x) = (f (x −− ))−− . Theorem 23. (a) If (M, v) is an Rvt -monoid then the mapping v˜ satisfies conditions (1), (2 ) and (3) from the definition of a near vt-operator. (b) If M satisfies the identity (x ∨ y)−− = x −− ∨ y −− , then v˜ is a near vt-operator on M. Proof. (a) We will show that v˜ satisfies conditions (1), (2 ) and (3): (1) v(1) ˜ = (v(1−− ))−− = 1.
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−− . (2 ) v(x −− )x −− ⇒ (v(x −− ))−− x −− ⇒ v(x)x ˜ −− −− −− −− = (v(x −− ) → v(y −− ))−− (v(x −− → y −− ))−− = (v((x → → (v(y )) (3) v(x) ˜ → v(y) ˜ = (v(x )) −− −− y) )) = v(x ˜ → y),
therefore v(x ˜ → y) v(x) ˜ → v(y). ˜ (b) Let M satisfy (x ∨ y)−− = x −− ∨ y −− for each x, y ∈ M. Then by (4) from the definition of a vt-operator, v(x) ˜ ∨ v(y) ˜ = (v(x −− ))−− ∨ (v(y −− ))−− = (v(x −− ) ∨ (v(y −− ))−− (v(x −− ∨ y −− ))−− = (v((x ∨ y)−− ))−− = v(x ˜ ∨ y).
Remark 24. (a) Every BL-algebra satisfies the identity (x ∨ y)−− = x −− ∨ y −− [5, Corollary 3.21]. (b) Let w be any near vt-operator on an R-monoid M satisfying the identity (x ∨ y)−− = x −− ∨ y −− . Then w(x) ˜ = (w(x −− ))−− (x −−−− )−− = x −− , thus w˜ satisfies condition (2 ). Moreover, from properties (1), (3) and (4) of w we get (analogously as in Theorem 23) that w˜ also satisfies conditions (1), (3) and (4), hence w˜ is also a near vt-operator. Now, let us denote by N VT (M) the set of all near vt-operators on an R-monoid M. (In the following assertions we will suppose that M satisfies the identity (x ∨ y)−− = x −− ∨ y −− .) If w1 , w2 ∈ N VT (M), we put w1 w2 if and only if w1 (x)w2 (x) for every x ∈ M. Then (N VT (M), ) is an (pointwise) ordered set. Let us consider the ˜ for every w ∈ N VT (M). mapping ∼ : N VT (M) −→ N VT (M) such that ∼ : w → w, Lemma 25. The mapping “∼ ” is a closure operator on the ordered set (N VT (M), ). Proof. Let w, w1 , w2 ∈ N VT (M). (1) (2) (3)
w(x) ˜ = ((w(x −− ))−− (w(x))−− w(x). Let w1 w2 . Then w1 (x −− ) w2 (x −− ) ⇒ (w1 (x −− ))−− (w2 (x −− ))−− ⇒ w˜1 (x) w˜2 (x), hence w˜1 w˜2 . ˜˜ ˜ w(x) = (w(x ˜ −− ))−− = ((w(x −−−− ))−− )−− = (w(x −− ))−− = w(x).
Definition 26. We say that a near vt-operator on M is essential if w is a closed element in (N VT (M), ) with respect to the closure operator “∼ ”. Proposition 27. If w ∈ N VT (M) then w and w˜ induce the same operators w − and (w) ˜ −. Proof. If x ∈ M then (w) ˜ − (x) = (w(x ˜ − ))− = ((w(x −−− ))−− )− = (w(x − ))− = w− (x).
The following proposition is an immediate consequence. Proposition 28. If M is an MV-algebra then every near vt-operator on M is essential. We will finish the paper by examples of operators on an R-monoid which is not a BL-algebra. As it was mentioned, an R-monoid M is isomorphic to a subdirect product of linearly ordered R-monoids (= linearly ordered BL-algebras) if and only if M is a BL-algebra. But R-monoids can be constructed from BL-algebras by means of other natural operations, e.g. by means of pastings (= ordinal sums). Note that by [18,19], the pasting of linearly ordered Wajsberg algebras is a linearly ordered BL-algebra, but the pasting of Wajsberg algebras which are not linear, gives bounded commutative R-monoids which are not BL-algebras. (Recall that the results from [18] concerning the ordinal sums were generalized in [20] to bounded BCK algebras and in [21] to bounded pseudo-BCK algebras.) Now, we will deal with operators on the R-monoid which is the pasting of four and two element MV-algebras (and is not a BL-algebra). Example 29. Let M = {0, a, b, c, 1} be the lattice with the diagram in Fig. 1, and let = ∧ and → is defined in the corresponding table in Fig. 1.
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1
→
0
a
b
c
1
0
1
1
1
1
1
a
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1
b
1
1
b
a
a
1
1
1
c
0
a
b
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1
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0
a
b
c
1
c
a
b
0 Fig. 1.
By [19], M is a pasting of the four element Wajsberg algebra {0, a, b, c} and the two element Wajsberg algebra {c, 1} and M is not a BL-algebra (since (a → b) ∨ (b → a) = b ∨ a = c = 1). Let v1 : M −→ M, v2 : M −→ M and v3 : M −→ M be the mappings such that (a) v1 (0) = v1 (a) = 0, v1 (b) = v1 (c) = b, v1 (1) = 1; (b) v2 (0) = v2 (a) = 0, v2 (b) = b, v2 (c) = c, v2 (1) = 1; (c) v3 (0) = v3 (a) = 0, v3 (b) = b, v3 (c) = v3 (1) = 1. Then we have (a) v1 is a vt-operator on M; (b) v2 is a wvt-operator on M, which is not a vt-operator because v2 (a ∨ b) = v2 (c) = c, v2 (a) ∨ v2 (b) = 0 ∨ b = b, but cb; (c) v3 is a mapping satisfying conditions (1) and (3) from the definition of a wvt-operator, not satisfying condition (2) because 1 = v3 (c)c, but satisfying condition (2 ) because v3 (c) c−− = (c → 0) → 0 = 1. All vi have the same vi− , (i = 1, 2, 3) : vi− (0) = 0, vi− (a) = a, vi− (b) = 1, vi− (c) = 1, vi− (1) = 1. Moreover, in the (pointwise) ordered set of mappings satisfying conditions (1), (2 ) and (3) we have v1 < v2 < v3 and v˜1 = v˜2 = v˜3 = v3 . Acknowledgement The authors are very indebted to the anonymous reviewers for their valuable comments and suggestions which helped to improve the paper. References [2] R. Balbes, P. Dwinger, Distributive Lattices, University of Missouri Press, 1974. [3] K. Blount, C. Tsinakis, The structure of residuated lattices, Internat. J. Algebra Comput. 13 (2003) 437–461. [4] R.L.O. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundation of Many-valued Reasoning, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000. [5] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: part I, Multi. Val. Logic 8 (2002) 673–714. [6] A. Di Nola, S. Sessa, F. Esteva, L. Godo, P. Garcia, The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting, Ann. Math. Artificial Intelligence 35 (2002) 197–214. [8] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288.
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[9] F. Esteva, L. Godo, A. Garcia-Cerdana, On the hierarchy of t-norm based residuated fuzzy logics, preprint. [10] J.M. Font, A.J. Rodriguez, A. Torrens, Wajsberg algebras, Stochastica 8 (1984) 5–31. [11] S. Gottwald, Fuzzy set theory with t-norms and -operators, in: A. Di Nola, A.G.S. Ventre (Eds.), The Mathematics of Fuzzy Systems, Interdisciplinary Syst. Res. 88 (1986) 143–195. [12] S. Gottwald, Fuzzy Sets and Fuzzy Logic, Braunschweig, Wiesbaden, Vieweg, 1993. [13] S. Gottwald, A Treatise on Many-Valued Logics, Studies in Logic and Computation, vol. 9, Research Studies Press, Baldock, Hertfordshire, England, 2001. [15] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Amsterdam, 1998. [16] P. Hájek, On very true, Fuzzy Sets and Systems 124 (2001) 329–333. [17] U. Höhle, Commutative, residuated l-monoids, in: U. Höhle, E.P. Klement (Eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1995, pp. 53–106. [18] A. Iorgulescu, Classes of BCK Algebras—Part I, Preprint Series of the Institute of Mathematics of the Romanian Academy, preprint nr. 1/2004, pp. 1–33. [19] A. Iorgulescu, Classes of BCK Algebras—Part III, Preprint Series of the Institute of Mathematics of the Romanian Academy, preprint nr. 3/2004, pp. 1–37. [20] A. Iorgulescu, On BCK Algebras—Part I, submitted for publication. [21] A. Iorgulescu, Classes of pseudo-BCK Algebras—part II, J. Multiple-Valued Logic Soft Comput., to appear. [22] P. Jipsen, C. Tsinakis, A survey of residuated lattices, in: J. Martinez (Ed.), Ordered Algebraic Structures, Kluwer Academic Publishers, Dordrecht, 2002, pp. 19–56. [23] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000. [24] T. Kowalski, H. Ono, Residuated lattices: an algebraic glimpse at logics without contraction, Monograph, 2001. [25] J. Kühr, Pseudo BL-algebras and DR-monoids, Math. Bohem. 128 (2003) 199–208. [26] J. Rach˚unek, A duality between algebras of basic logic and bounded representable DR-monoids, Math. Bohem. 126 (2001) 561–569. [27] J. Rach˚unek, D. Šalounová, Local bounded commutative residuated -monoids, Czechoslovak Math. J., to appear. [28] J. Rach˚unek, V. Slezák, Negation in bounded commutative DR-monoids, Czechoslovak Math. J. 56 (2006) 755–763. [30] K.L.N. Swamy, Dually residuated lattice ordered semigroups, Math. Ann. 159 (1965) 105–114. [31] V. Vychodil, Truth-depressing hedges and BL-logic, Fuzzy Sets and Systems 157 (2006) 2074–2090. [32] M. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939) 335–354. [33] L. Zadeh, Fuzzy logic and approximate reasoning, Synthese 30 (1975) 407–428.
Truth values on generalizations of some commutative fuzzy structures Jiˇrí Rach˚uneka,∗,1 , Dana Šalounováb a Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic b Department of Mathematical Methods in Economy, Faculty of Economics, VŠB—Technical University Ostrava, Sokolská 33,
701 21 Ostrava, Czech Republic Received 1 December 2005; received in revised form 24 August 2006; accepted 30 August 2006 Available online 25 September 2006
Abstract Hájek introduced the logic BLvt enriching the logic BL by a unary connective vt which is a formalization of Zadeh’s fuzzy truth value “very true”. BLvt -algebras, i.e. BL-algebras with unary operations, called vt-operators, which are among others subdiagonal, are an algebraic counterpart of BLvt . Residuated lattice ordered monoids (R-monoids) are common generalizations of BL-algebras and Heyting algebras. In the paper, we study algebraic properties of Rvt -algebras (and consequently of BLvt -algebras) and of those that are enriched by derived superdiagonal operators which in the case of MV-algebras are the duals to vt-operators. © 2006 Elsevier B.V. All rights reserved. MSC: 03B47; 03B52; 03G25; 06D35; 06F05 Keywords: Residuated -monoid; Basic fuzzy logic; BL-algebra; Fuzzy truth value
1. Introduction Inspired by the considerations of Zadeh [33], Hájek in [16] formalized the fuzzy truth value “very true”. He enriched the language of the basic fuzzy logic BL by adding a new unary connective vt and introduced the (propositional) logic BL vt . The completeness of BL vt was proved in [16] by using the so-called BL vt -algebras, an algebraic counterpart of BL vt . Recall that a BL vt -algebra is a BL-algebra with a unary operation (an operator) which is, among others, subdiagonal. Bounded commutative residuated lattice ordered monoids (R-monoids) are algebraic structures which generalize, e.g. both BL-algebras and Heyting algebras (an algebraic counterpart of the intuitionistic propositional logic). Nevertheless, many of properties of BL-algebras are also satisfied in any bounded commutative R-monoids. Therefore, bounded commutative R-monoids could be taken as an algebraic semantics of a more general logic than Hájek’s fuzzy logic. It is known that every BL-algebra (and consequently every MV-algebra) can be represented as a subdirect product of linearly ordered BL-algebras. (Recall also that MV-algebras are categorically isomorphic to Wajsberg algebras [10].) ∗ Corresponding author. Tel.: +420 585 634 650; fax: +420 585 634 663.
E-mail addresses: [email protected] (J. Rach˚unek), [email protected] (D. Šalounová). 1 Supported by the Council of Czech Government, MSM 6198959214.
0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.08.007
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It is also true [26] that a bounded commutative R-monoid is a subdirect product of linearly ordered R-monoids if and only if it is a BL-algebra. On the other side, R-monoids which need not be BL-algebras can be constructed from BL-algebras by means of other natural operations, e.g. by means of pastings (= ordinal sums). Note that by [18,19], the pasting of linearly ordered Wajsberg algebras is a linearly ordered BL-algebra, but the pasting of Wajsberg algebras which are not linear, gives bounded commutative R-monoids which are not BL-algebras. Therefore, by a “commutative fuzzy structure” we mean here any BL-algebra and as its appropriate generalization is considered any bounded commutative R-monoid. In this paper we introduce Rvt -monoids as bounded commutative R-monoids with a unary subdiagonal and monotone self-mapping (called a vt-operator) which generalize BL vt -algebras. Further, we derive from the vt-operators, in a purely algebraic and uniform way, superdiagonal and monotone self-mappings which are in the case of MV-algebras vf -operators, i.e. formalizations of the fuzzy truth value “very false”. Note that in the case of BL-algebras, vt-operators induce so-called st-operators that have been introduced very recently by Vychodil [31] as formalization of the fuzzy truth value “slightly true”. Note, nevertheless, that these superdiagonal operators are the duals to vt-operators only in the case of MV-algebras. For concepts and results concerning MV-algebras, BL-algebras and Heyting algebras see for instance [4,15,2]. 2. Bounded commutative R-monoids Bounded commutative R-monoids form a large class of algebras containing as proper subclasses, among others, the classes of algebras of some logics, e.g. the class of BL-algebras, i.e. algebras of the basic fuzzy logic (and consequently the class of MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic), as well as the class of Heyting algebras, i.e. algebras of the intuitionistic logic. Bounded commutative R-monoids have many common properties of both mentioned kinds of algebras of logics and hence, they could be taken as the algebras of a logic which generalizes fuzzy logics and admits also the intuitionistic principles. Definition 1. A bounded commutative R-monoid is an algebra M = (M; ∨, ∧, , →, 0, 1) of type 2, 2, 2, 2, 0, 0 satisfying the following conditions: (i) (ii) (iii) (iv)
(M; , 1) is a commutative monoid. (M; ∨, ∧, 0, 1) is a bounded lattice. x y z if and only if x y → z for any x, y, z ∈ M. x (x → y) = x ∧ y for any x, y ∈ M.
Remark 2. In fact, the notion of a bounded commutative R-monoid is a duplicate name for a commutative residuated lattice [32,24] satisfying divisibility condition or for a divisible commutative residuated lattice [18], which is categorically isomorphic to a divisible BCK(P) lattice [18], or for a divisible integral residuated commutative -monoid [17] or for a bounded commutative integral generalized BL-algebra [3,22]. In the sequel, by an R-monoid we will mean a bounded commutative R-monoid. It is possible to show that for any R-monoid M, the lattice (M; ∨, ∧) is distributive and that operation distributes over the lattice operations ∨ and ∧. Let us define on any R-monoid M the unary operation − (negation) by x − := x → 0. Algebras of the above mentioned propositional logics can be characterized in the class of all R-monoids as follows: An R-monoid M is (a) a BL-algebra [26] if and only if M satisfies the identity of pre-linearity (x → y) ∨ (y → x) = 1; (b) an MV-algebra [17], see also [9], if and only if M fulfils the double negation law x −− = x; (c) a Heyting algebra [30] if and only if the operations and ∧ coincide on M.
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Remark 3. Let us recall that commutative residuated lattices which satisfy the identity of pre-linearity are also called MTL-algebras and by [8,6] they are an algebraic counterpart of the so-called Monoidal t-norm logic (MTL, for short). Now it is obvious that an R-monoid is an MTL-algebra if and only if it is a BL-algebra. The facts in the rest of the section can be verified as for the BL case, since the pre-linearity condition is not involved, and hence the proofs are omitted. Lemma 4. In any bounded commutative R-monoid M we have for any x, y, z ∈ M: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
x y ⇐⇒ x → y = 1. x y ⇒ x z y z. x y ⇒ z → x z → y, y → z x → z. x → x = 1, 1 → x = x, x → 1 = 1. y x → y. x x −− , x − = x −−− . x y ⇒ y − x − . (x ∨ y)− = x − ∧ y − . (x ∧ y)−− = x −− ∧ y −− . (x y)− = y → x − = y −− → x − = x → y − = x −− → y − . (x → y)−− = x −− → y −− .
If M is an R-monoid and ∅ = F ⊆ M, then F is called a filter of M if (i) x, y ∈ F ⇒ x y ∈ F ; (ii) x ∈ F, y ∈ M, x y ⇒ y ∈ F . The filters of R-monoids coincide with the kernels of their congruences. If F is a filter of M then F is the kernel of the unique congruence (F ) such that x, y ∈ (F ) if and only if (x → y) ∧ (y → x) ∈ F , for any x, y ∈ M. Hence, we will consider quotient R-monoids M/F of R-monoid M with respect to its filters F . 3. R-monoids with vt-operators In this section we deal with Rvt -monoids which are generalizations of BL vt -algebras introduced in [16]. Definition 5. (a) Let M be an R-monoid. A mapping v : M −→ M is called a weak vt-operator (wvt-operator in brief) on M if for any x, y ∈ M: (1) v(1) = 1, (2) v(x)x, i.e. v is subdiagonal, (3) v(x → y) v(x) → v(y). (b) If a wvt-operator v satisfies for any x, y ∈ M (4) v(x ∨ y)v(x) ∨ v(y), then v is called a vt-operator on M. Remark 6. Any R-monoid admits vt-operators, e.g. the identity and the globalization g, where g(x) = 0 for x = 1 and g(1) = 1. Lemma 7. Let v be a weak vt-operator on an R-monoid and x, y, z ∈ M. Then (a) (b) (c) (d)
v(0) = 0, x y ⇒ v(x)v(y), v(x − )(v(x))− , x y z ⇒ v(x) v(y)v(z),
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(e) v(x) v(y)v(x y), (f) v(x) v(x → y)v(x ∧ y)v(x) ∧ v(y). If v is a vt-operator on M and x, y ∈ M then (g) v(x ∨ y) = v(x) ∨ v(y). Proof. (a) By the definition, v(0) 0, hence v(0) = 0. (b) Let x, y ∈ M and x y. Then x → y = 1, hence by (3) and (1) from definition, we get v(x) → v(y) = 1, and thus v(x)v(y). (c) Let x ∈ M. Then by (3) from definition and by (a), v(x − ) = v(x → 0) v(x) → v(0) = v(x) → 0 = (v(x))− . (d) Let x y z. Then x y → z, so by (b) and (3), v(x) v(y → z) v(y) → v(z), and from this, v(x) v(y)v(z). (e) It follows from (d) for z = x y. (f) By (e) and (b), v(x) v(x → y)v(x (x → y)) = v(x ∧ y) v(x) ∧ v(y). (g) By (b), we have v(x) ∨ v(y)v(x ∨ y), hence by (4), v(x ∨ y) = v(x) ∨ v(y). Definition 8. If M is an R-monoid and v is a vt-operator (a wvt-operator) on M then the ordered pair (M, v) is called an Rvt -monoid (a Rwvt -monoid, respectively). Remark 9. (a) For the sake of simplicity, we will mainly study properties of Rvt -monoids and some operators derived from them. (b) If (M, v) is an Rvt -monoid and M is a BL-algebra then (M, v) is a BL vt -algebra in the sense of [16]. Definition 10. If (M, v) is an Rvt -monoid and F is a filter of M then F is called a v-filter of the Rvt -monoid (M, v) if v(x) ∈ F for every x ∈ F . For any v-filter F of an Rvt -monoid (M, v) denote by vF : M/F −→ M/F the mapping such that vF (x/F ) := v(x)/F, for each x ∈ M. Proposition 11. If F is a v-filter of an Rvt -monoid (M, v) then vF is a vt-operator on the quotient Rmonoid M/F . Proof. Firstly we will show that vF is a correctly defined mapping of M/F into M/F . Let x, y ∈ M and x/F = y/F . Then x, y ∈ (F ), i.e. (x → y) ∧ (y → x) ∈ F, and thus also x → y, y → x ∈ F. Since F is a v-filter, we get v(x → y), v(y → x) ∈ F, and hence by (3) from the definition of a vt-operator we obtain v(x) → v(y), v(y) → v(x) ∈ F. By Lemma 4(5), v(y) → v(x)(v(x) → v(y)) → (v(y) → v(x)), hence (v(x) → v(y)) → (v(y) → v(x)) ∈ F, and this means v(x), v(y) ∈ (F ), i.e. vF (x/F ) = vF (y/F ). Now it is easy to verify that the mapping vF is a vt-operator on M/F. (1) vF (1/F ) = v(1)/F = 1/F. (2) vF (x/F ) = v(x)/F x/F. (3) vF (x/F → y/F ) = vF ((x → y)/F ) = (v(x → y))/F (v(x) → v(y))/F = v(x)/F → v(y)/F = vF (x/F ) → vF (y/F ). (4) vF (x/F ∨ y/F ) = vF ((x ∨ y)/F ) = v(x ∨ y)/F (v(x) ∨ v(y))/F = (v(x)/F ) ∨ (v(y)/F ) = vF (x/F ) ∨ vF (y/F ). Definition 12. If (M, v) is an Rvt -monoid and is a congruence on the R-monoid M then is called a v-congruence on (M, v), if x, y ∈ implies v(x), v(y) ∈ for each x, y ∈ M. Proposition 13. If (M, v) is an Rvt -monoid then there is a one-to-one correspondence between its v-filters and v-congruences.
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Proof. (a) Let be a v-congruence on (M, v) and let F = 1/F = {x ∈ M : x, 1 ∈ }. Then F is a filter of the R-monoid M. Let us suppose that x ∈ F . Then x, 1 ∈ , hence v(x), 1 = v(x), v(1) ∈ , and therefore v(x) ∈ F . That means F is a v-filter on (M, v). (b) Let F be a v-filter of (M, v) and let F be the corresponding congruence on M, i.e. x, y ∈ F if and only if (x → y) ∧ (y → x) ∈ F. Hence, if x, y ∈ F then also v((x → y) ∧ (y → x)) ∈ F . Let x, y ∈ F . Then by property (3) of a vt-operator and Lemma 7(f), (v(x) → v(y)) ∧ ((v(y) → v(x))v(x → y) ∧ v(y → x)v((x → y) ∧ (y → x)) ∈ F, hence (v(x) → v(y)) ∧ ((v(y) → v(x)) ∈ F, and this means v(x), v(y) ∈ F . Therefore F is a v-congruence on (M, v). Now we will deal with Rwvt -monoids (M, v) satisfying the identity (5) v(x → y) ∨ v(y → x) = 1. Theorem 14. If M is an R-monoid then there is a wvt-operator v on M satisfying (5) if and only if M is a BL-algebra. Proof. (a) Let M be an R-monoid which is not a BL-algebra. Then there exist x, y ∈ M such that (x → y) ∨ (y → x) = 1. Hence for any wvt-operator v on M, v(x → y) ∨ v(y → x)(x → y) ∨ (y → x) < 1, therefore (5) fails. (b) Let M be a BL-algebra. Then for each vt-operator v on M and x, y ∈ M, v(x → y) ∨ v(y → x) = v((x → y) ∨ (y → x)) = v(1) = 1, thus M satisfies (5).
An Rvt -monoid (M, v) is called an Rvt -chain if the R-monoid M is linearly ordered. By [26], the class of BLalgebras coincides with the class of (bounded commutative) R-monoids which are representable as subdirect products of R-chains. Hence, among others, every Rvt -chain is in fact a BL vt -chain. We will prove that every BL vt -algebra is a subdirect product not only of BL-chains (i.e., as a BL-algebra in the corresponding signature), but, moreover, it is also such a product of BL vt -chains (in the extended signature). Recall that a filter P of an R-monoid M is called prime if P = F ∩ G implies P = F or P = G for any filters F and G of M. A prime filter is called minimal if it is a minimal element in the sets of prime filters of M ordered by set inclusion. By Zorn’s lemma, every prime filter of M contains a minimal prime filter. For any a ∈ M put a ⊥ = {x ∈ M : x ∨ a = 1}. If P is a prime filter of M then x ∨ y = 1 implies x ∈ P or y ∈ P for each x, y ∈ M and then the quotient R-monoid M/P is linearly ordered. If P is a minimal prime filter of M then P = {a ⊥ : a ∈ M \ P }. (See e.g. [25].) Theorem 15. Every BL vt -algebra is a subdirect product of BL vt -chains. Proof. It is obvious that it suffices to prove that every BL vt -algebra is isomorphic to subdirect product of BL vt -chains. Since any BL-algebra M is representable as a subdirect product of BL-chains, the intersection of all minimal prime filters of M is equal to {1}. Hence, it remains to show that every minimal prime filter of M is a v-filter. Let P be a minimal prime filter of M. Then P = {a ⊥ : a ∈ M \ P }. Let x ∈ P . Then there exists a ∈ / P such that x ∨ a = 1, hence 1 = v(1) = v(x ∨ a) = v(x) ∨ v(a). Since a ∈ / P , we get v(a) ∈ / P , therefore (P is a prime filter) v(x) ∈ P . 4. Operators on R-monoids derived from vt-operators Every vt-operator on an R-monoid M is, by the definition and Lemma 7(b), a subdiagonal and monotone selfmapping of M. Now, we use vt-operators to introduce derived self-mappings of M that are, among others, superdiagonal and monotone, and in the case of MV-algebras they have the properties of unary connectives “very false”.
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If M is a R-monoid and f : M −→ M then we denote by f − the mapping of M into M such that for any x ∈ M, f − (x) = (f (x − ))− . Example 16. Let us consider the standard MV-algebra [0, 1] = (R, 1). It is known that the mapping v : [0, 1] − → [0, 1] such that v(x) = x 2 is a vt-operator on [0, 1]. Then v − : [0, 1] −→ [0, 1] is the mapping such that v − (x) = 2x − x 2 for each x ∈ [0, 1]. We say that an R-monoid M is normal if M satisfies the identity (x y)−− = x −− y −− . Remark 17. Every BL-algebra and every Heyting algebra is normal (see [28]), hence the variety of normal R-monoids is considerably wide. Proposition 18. Let (M, v) be an Rvt -monoid. Then we have for any x, y, z ∈ M: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
v − (0) = 0, v − (1) = 1, x v − (x), x y implies v − (x)v − (y), v − (x ∧ y) v − (x) ∧ v − (y), v − (x ∨ y) v − (x) ∨ v − (y), v − (x → y −− )v(x) → v − (y), x y z implies v − (x) v(y)v − (z), v − (x) v(y)v − (x y), v − (x) v(x → y) v − (x ∧ y). If M is normal then v − (x → y)v(x −− ) → v − (y) v(x) → v − (y). If M is an MV-algebra then v − (x → y)v(x) → v − (y).
Proof. 1. v − (0) = (v(0− ))− = 1− = 0, v − (1) = (v(1− ))− = (v(0))− = 0− = 1. 2. v − (x) = ((v(x − ))− x −− x. 3. x y ⇒ x − y − ⇒ v(x − ) v(y − ) ⇒ (v(x − ))− (v(y − ))− ⇒ v − (x) v − (y). 4. and 5. They follow from 3. 6. Using Lemma 4(6), (7), (12) and Lemma 7(a), (c), we have v − (x → y −− ) = v − ((xy − )− ) = (v((xy − )−− ))− , v(x) → v − (y) = v(x) → (v(y − ))− = ((v(x) v(y − ))− (v(x y − ))− (v((x y − )−− )− , hence v − (x → y −− )v(x) → v − (y). 7. By 3 and 6 and by Lemma 4(3) we get x y z ⇒ x y → z ⇒ v − (x)v − (y → z)v − (y → z−− ) ⇒ v − (x)v(y) → v − (z) ⇒ v − (x) v(y) v − (z). 8. It follows from 7. 9. v − (x) v(x → y)v − (x (x → y)) = v − (x ∧ y). 10. By [27, Lemma 2.3], x → y y − → x − , thus from the normality of M we get v − (x → y) = (v((x → y)− ))− (v((y − → x − )− ))− = (v((y − x)−− ))− = (v(y − x −− ))− . Furthermore, (v(y − ) v(x −− ))− = v(x −− ) → (v(y − ))− = v(x −− ) → v − (y), and since by Lemma 7(d), (v(y − x −− ))− (v(y − ) v(x −− ))− , we obtain v − (x → y) v(x −− ) → v − (y). 11. It follows from 6 as well as from 10. If M is an MV-algebra then M satisfies the double negation law x −− = x, hence there exist on M both a t-norm and its associated residuum → and a t-conorm and its associated residuum, say . (See [11–13,23,18].)
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Consequently, on any MV-algebra M one can define not only vt-operators but also dual operators, vf -operators (vf = very false). We will show that every vt-operator on M determines a vf -operator on M.As usual, put x y := (x − y − )− and x y := x y − , for any x, y ∈ M. Proposition 19. If M is an MV-algebra and v is a vt-operator on M then v − is a vf -operator on M, i.e. for each x, y ∈ M it holds: (1− ) (2− ) (3− ) (4− )
v − (0) = 0, v − (x)x, v − (x) v − (y)v − (x y), v − (x ∧ y) = v − (x) ∧ v − (y).
Proof. It remains to verify properties (3− ) and (4− ). (3− ): Using 8 from Proposition 18, we get v − (x) v − (y) = (v(x − )− (v(y − ))− = (v(x − ))− (v(y − ))−− = (v(x − ))− v(y − ) = v − (x) v(y − ) v − (x y − ) = v − (x y). (4− ): Since any MV-algebra satisfies de Morgan laws for the lattice operations, we have v − (x ∧ y) = (v((x ∧ y)− ))− = (v(x − ∨ y − ))− = (v(x − ) ∨ v(y − ))− = (v(x − ))− ∧ (v(y − ))− = v − (x) ∧ v − (y). Remark 20. Very recently, Vychodil [31] has introduced the notion of a BL vt,st -algebra in order to study the socalled truth depressing hedges on BL-algebras. An algebra (M; ∨, ∧, , →, v, s, 0, 1) is called a BL vt,st -algebra if M = (M; ∨, ∧, , →, 0, 1) is a BL-algebra, v is a wvt-operator on M and s : M −→ M is a mapping such that () s(0) = 0; () x s(x); () v(x → y)s(x) → s(y). We will show that if M is a BL-algebra and v is a wvt-operator on M then (M; ∨, ∧, , →, v, v − , 0, 1) is a BL vt,st algebra. Let x, y ∈ M. Then, by Lemma 4(8), v − (x) → v − (y) = (v − (x − ))− → (v(y − ))− v(y − ) → v(x − ) v(y − → x − )v(x → y). Definition 21. Let M be an R-monoid. A mapping w : M −→ M is called a near vt-operator on M if it satisfies conditions (1), (3) and (4) from the definition of a vt-operator on M and if for any x ∈ M, (2 ) w(x)x −− . Remark 22. By Lemma 4(6), every vt-operator on M is a near vt-operator on M. Let M be an R-monoid and f : M −→ M be a mapping. Let us denote by f˜ the self-mapping of M such that ˜ f = (f − )− , i.e., for each x ∈ M, f˜(x) = (f (x −− ))−− . Theorem 23. (a) If (M, v) is an Rvt -monoid then the mapping v˜ satisfies conditions (1), (2 ) and (3) from the definition of a near vt-operator. (b) If M satisfies the identity (x ∨ y)−− = x −− ∨ y −− , then v˜ is a near vt-operator on M. Proof. (a) We will show that v˜ satisfies conditions (1), (2 ) and (3): (1) v(1) ˜ = (v(1−− ))−− = 1.
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−− . (2 ) v(x −− )x −− ⇒ (v(x −− ))−− x −− ⇒ v(x)x ˜ −− −− −− −− = (v(x −− ) → v(y −− ))−− (v(x −− → y −− ))−− = (v((x → → (v(y )) (3) v(x) ˜ → v(y) ˜ = (v(x )) −− −− y) )) = v(x ˜ → y),
therefore v(x ˜ → y) v(x) ˜ → v(y). ˜ (b) Let M satisfy (x ∨ y)−− = x −− ∨ y −− for each x, y ∈ M. Then by (4) from the definition of a vt-operator, v(x) ˜ ∨ v(y) ˜ = (v(x −− ))−− ∨ (v(y −− ))−− = (v(x −− ) ∨ (v(y −− ))−− (v(x −− ∨ y −− ))−− = (v((x ∨ y)−− ))−− = v(x ˜ ∨ y).
Remark 24. (a) Every BL-algebra satisfies the identity (x ∨ y)−− = x −− ∨ y −− [5, Corollary 3.21]. (b) Let w be any near vt-operator on an R-monoid M satisfying the identity (x ∨ y)−− = x −− ∨ y −− . Then w(x) ˜ = (w(x −− ))−− (x −−−− )−− = x −− , thus w˜ satisfies condition (2 ). Moreover, from properties (1), (3) and (4) of w we get (analogously as in Theorem 23) that w˜ also satisfies conditions (1), (3) and (4), hence w˜ is also a near vt-operator. Now, let us denote by N VT (M) the set of all near vt-operators on an R-monoid M. (In the following assertions we will suppose that M satisfies the identity (x ∨ y)−− = x −− ∨ y −− .) If w1 , w2 ∈ N VT (M), we put w1 w2 if and only if w1 (x)w2 (x) for every x ∈ M. Then (N VT (M), ) is an (pointwise) ordered set. Let us consider the ˜ for every w ∈ N VT (M). mapping ∼ : N VT (M) −→ N VT (M) such that ∼ : w → w, Lemma 25. The mapping “∼ ” is a closure operator on the ordered set (N VT (M), ). Proof. Let w, w1 , w2 ∈ N VT (M). (1) (2) (3)
w(x) ˜ = ((w(x −− ))−− (w(x))−− w(x). Let w1 w2 . Then w1 (x −− ) w2 (x −− ) ⇒ (w1 (x −− ))−− (w2 (x −− ))−− ⇒ w˜1 (x) w˜2 (x), hence w˜1 w˜2 . ˜˜ ˜ w(x) = (w(x ˜ −− ))−− = ((w(x −−−− ))−− )−− = (w(x −− ))−− = w(x).
Definition 26. We say that a near vt-operator on M is essential if w is a closed element in (N VT (M), ) with respect to the closure operator “∼ ”. Proposition 27. If w ∈ N VT (M) then w and w˜ induce the same operators w − and (w) ˜ −. Proof. If x ∈ M then (w) ˜ − (x) = (w(x ˜ − ))− = ((w(x −−− ))−− )− = (w(x − ))− = w− (x).
The following proposition is an immediate consequence. Proposition 28. If M is an MV-algebra then every near vt-operator on M is essential. We will finish the paper by examples of operators on an R-monoid which is not a BL-algebra. As it was mentioned, an R-monoid M is isomorphic to a subdirect product of linearly ordered R-monoids (= linearly ordered BL-algebras) if and only if M is a BL-algebra. But R-monoids can be constructed from BL-algebras by means of other natural operations, e.g. by means of pastings (= ordinal sums). Note that by [18,19], the pasting of linearly ordered Wajsberg algebras is a linearly ordered BL-algebra, but the pasting of Wajsberg algebras which are not linear, gives bounded commutative R-monoids which are not BL-algebras. (Recall that the results from [18] concerning the ordinal sums were generalized in [20] to bounded BCK algebras and in [21] to bounded pseudo-BCK algebras.) Now, we will deal with operators on the R-monoid which is the pasting of four and two element MV-algebras (and is not a BL-algebra). Example 29. Let M = {0, a, b, c, 1} be the lattice with the diagram in Fig. 1, and let = ∧ and → is defined in the corresponding table in Fig. 1.
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1
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By [19], M is a pasting of the four element Wajsberg algebra {0, a, b, c} and the two element Wajsberg algebra {c, 1} and M is not a BL-algebra (since (a → b) ∨ (b → a) = b ∨ a = c = 1). Let v1 : M −→ M, v2 : M −→ M and v3 : M −→ M be the mappings such that (a) v1 (0) = v1 (a) = 0, v1 (b) = v1 (c) = b, v1 (1) = 1; (b) v2 (0) = v2 (a) = 0, v2 (b) = b, v2 (c) = c, v2 (1) = 1; (c) v3 (0) = v3 (a) = 0, v3 (b) = b, v3 (c) = v3 (1) = 1. Then we have (a) v1 is a vt-operator on M; (b) v2 is a wvt-operator on M, which is not a vt-operator because v2 (a ∨ b) = v2 (c) = c, v2 (a) ∨ v2 (b) = 0 ∨ b = b, but cb; (c) v3 is a mapping satisfying conditions (1) and (3) from the definition of a wvt-operator, not satisfying condition (2) because 1 = v3 (c)c, but satisfying condition (2 ) because v3 (c) c−− = (c → 0) → 0 = 1. All vi have the same vi− , (i = 1, 2, 3) : vi− (0) = 0, vi− (a) = a, vi− (b) = 1, vi− (c) = 1, vi− (1) = 1. Moreover, in the (pointwise) ordered set of mappings satisfying conditions (1), (2 ) and (3) we have v1 < v2 < v3 and v˜1 = v˜2 = v˜3 = v3 . Acknowledgement The authors are very indebted to the anonymous reviewers for their valuable comments and suggestions which helped to improve the paper. References [2] R. Balbes, P. Dwinger, Distributive Lattices, University of Missouri Press, 1974. [3] K. Blount, C. Tsinakis, The structure of residuated lattices, Internat. J. Algebra Comput. 13 (2003) 437–461. [4] R.L.O. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundation of Many-valued Reasoning, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000. [5] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: part I, Multi. Val. Logic 8 (2002) 673–714. [6] A. Di Nola, S. Sessa, F. Esteva, L. Godo, P. Garcia, The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting, Ann. Math. Artificial Intelligence 35 (2002) 197–214. [8] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288.
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